A Spatial Approach to Network Generation for Three Properties: Degree Distribution, Clustering Coefficient and Degree Assortativity

Network topologies can be highly non-trivial, due to the complex underlying behaviours that form them. While past research has shown that some processes on networks may be characterized by local statistics describing nodes and their neighbours, such as degree assortativity, these quantities fail to capture important sources of variation in network structure. We define a property called transsortativity that describes correlations among a node's neighbours. Transsortativity can be systematically varied, independently of the network's degree distribution and assortativity. Moreover, it can significantly impact the spread of contagions as well as the perceptions of neighbours, known as the majority illusion. Our work improves our ability to create and analyse more realistic models of complex networks.

Cryptocurrency is a well-developed blockchain technology application that is currently a heated topic throughout the world. The public availability of transaction histories offers an opportunity to analyze and compare different cryptocurrencies. In this paper, we present a dynamic network analysis of three representative blockchain-based cryptocurrencies: Bitcoin, Ethereum, and Namecoin. By analyzing the accumulated network growth, we find that, unlike most other networks, these cryptocurrency networks do not always densify over time, and they are changing all the time with relatively low node and edge repetition ratios. Therefore, we then construct separate networks on a monthly basis, trace the changes of typical network characteristics (including degree distribution, degree assortativity, clustering coefficient, and the largest connected component) over time, and compare the three. We find that the degree distribution of these monthly transaction networks cannot be well fitted by the famous power-law distribution, at the same time, different currency still has different network properties, e.g., both Bitcoin and Ethereum networks are heavy-tailed with disassortative mixing, however, only the former can be treated as a small world. These network properties reflect the evolutionary characteristics and competitive power of these three cryptocurrencies and provide a foundation for future research.

Communities are virtually ubiquitous in real-world networks, and the statistic of modularity index Q is the classical measurement for community detection algorithms. However, the relationship between the modularity property and network multilever micro-scale structures is still not clear. In this paper, we study community detection results both in artificial and real-life complex networks by constructing different order null networks, and the results uncover that how micro-structures (such as degree distribution, assortativity and clustering coefficient) affect community properties. Meanwhile, we also propose two novel null networks (increasing or decreasing community structures) to verify the robustness of different community detection algorithms. Our results indicate that the modularity index Q is not a suitable statistic to measure the weak community property which is widely available in empirical networks. Our findings can not only be used to test the robustness of different community detection methods, but also be helpful to uncover the correlation of network structures between microcosmic and mesoscopic scales.

We present a detailed study of the phase diagram of the Ising model in random graphs with arbitrary degree distribution. By using the replica method we compute exactly the value of the critical temperature and the associated critical exponents as a function of the minimum and maximum degree, and the degree distribution characterizing the graph. As expected, there is a ferromagnetic transition provided <k> <= <k^2> < \infty. However, if the fourth moment of the degree distribution is not finite then non-trivial scaling exponents are obtained. These results are analyzed for the particular case of power-law distributed random graphs.

Importance:Early alcohol initiation (before age 15) is associated with adverse outcomes. Understanding mechanisms behind early alcohol initiation is essential for informing prevention efforts.Objective:To examine whether structural covariance network properties at ages 9–10 years predict early alcohol initiation.Design:Case-control, population-based study design.Setting:Data from the Adolescent Brain Cognitive Development study were used. Baseline structural brain imaging data (ages 9–10) were used for generation and comparison of structural covariance networks. Data from baseline to 4-year follow-up (≤age 15) assessments were used to determine alcohol initiation.Participants:Participants were excluded if they reported consuming a full drink of alcohol at baseline, or did not meet imaging inclusion criteria. Controls were excluded if they had not yet been assessed or were missing substance use data at 4-year follow-up. In total, 3,878 participants met study criteria, of which 182 participants initiated alcohol. Structural covariance network properties were compared between the full sample and a 1:1 propensity-matched sample based on age, sex, race, ethnicity, religion, parental education, prenatal alcohol exposure, and baseline alcohol sipping.Main Outcomes and Measures:Structural covariance networks were estimated using regional cortical thickness and volume measurements. Measures of network segregation (modularity, clustering coefficient), integration (characteristic path length, global efficiency), and resilience (degree assortativity) were compared between groups. Early alcohol initiation was defined as consuming a full drink between baseline and 4-year follow-upResults:Alcohol initiators (n=182, median[IQR] age, 10.3[9.9–10.8]; 101 female[55.5%]) demonstrated lower network segregation (modularity: area-under-the-curve[AUC] difference[95%CI]=-0.017[-0.017,-0.007], p=0.030; clustering coefficient: AUC[95%CI]=-0.026[-0.027,-0.012], p=0.0495) and higher network integration (characteristic path length: AUC[95%CI]=-0.106[-0.099,-0.046], p=0.020; global efficiency: AUC[95%CI]=0.011[0.005,0.011], p=0.010), compared to non-initiators (n=3,696, median[IQR] age, 9.9[9.4–10.4]; 1750 female[47.4%]) when controlling for age, sex, and mean cortical thickness. Within the matched sample, only differences in network integration were preserved (characteristic path length: AUC[95%CI]=-0.044[-0.032,0.035], p=0.010; global efficiency: AUC[95%CI]=0.003[-0.003,0.003], p=0.040). There were no differences between full or matched samples when comparing cortical volume structural covariance networks.Conclusions and Relevance:Differences in cortical thickness structural covariance network properties at ages 9–10 predicted alcohol initiation before age 15. These findings suggest cortical thickness network topology may reflect a neuroanatomical risk marker for early alcohol initiation.

Weibo is the Twitter counterpart in China that has attracted hundreds of millions of users. We crawled an almost complete Weibo user network that contains 222 million users and 27 billion links in 2013. This paper analyzes the structural properties of this network, and compares it with a Twitter user network. The topological properties we studied include the degree distributions, reciprocity, clustering coefficient, PageRank centrality, and degree assortativity. We find that Weibo users have a higher diversity index, higher Gini index, but a lower reciprocity and clustering coefficient for most of the nodes. A surprising observation is that the reciprocity of Weibo is only about a quarter of the reciprocity of the Twitter user network. We also show that Weibo adoption rate correlates with economic development positively, and Weibo network can be used to quantify the connections between provinces and regions in China. In particular, point-wise mutual information is shown to be accurate in quantifying the strength of connections.

Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fully-connected models, such as the Curie–Weiss, the Kuramoto, and the Sherrington–Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington–Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical findings and provide compelling evidence that the present mean-field theory describes an intermediate regime of connectivity, in which the average degree c scales as a power c ∝ N b (b < 1) of the total number N ≫ 1 of spins. Our findings put forward a novel class of spin models that incorporate the effects of degree fluctuations and, at the same time, are amenable to exact analytic solutions.

Emerging evidence suggests regional neuroanatomical variability may be predictive of early alcohol use (before age 15). However, the relationship between whole-brain network organization and early alcohol initiation remains unknown. Using data from the Adolescent Brain Cognitive Development (ABCD) study, we conducted a structural covariance network (SCN) analysis to examine brain network features preceding early alcohol initiation. Structural MRI data collected at baseline (ages 9–10) were used to generate SCNs based on regional cortical thickness measurements. Early alcohol initiation was defined as consuming a full drink between baseline and 4-year follow-up ( ≤ age 15). Participants who reported a full drink of alcohol at baseline, did not participate in the 4-year follow-up, or did not meet imaging quality control criteria were excluded. The remaining participants were compared to a subsample matched at a 1:1 ratio (n = 160 per group). SCN properties, including network segregation (modularity, clustering coefficient), integration (characteristic path length, global efficiency), and resilience (degree assortativity), were compared between groups. While no differences in regional cortical thickness between groups were identified, early initiators demonstrated lower segregation and higher integration compared to non-initiators. These findings suggest that cortical thickness network topology at ages 9–10 may serve as a neuroanatomical risk marker for early adolescent alcohol initiation, independent of prior alcohol exposure, sociodemographic differences, and regional neuroanatomical variability.

Properties of asymmetrical evolving networks

Research on growth of innovations introduced to the market has gradually shifted its focus from aggregate-level diffusion to exploring how growth is influenced by a given social network structure's characteristics. In this paper, we critically review this branch of literature. We argue that the growth of an innovation in a social network is shaped by the network's structure. Borrowing from the field of industrial organization in economics, which defines itself as the study of the effect of market structure on market performance, we describe this new wave of research on growth of innovations as the effect of social network structure on innovation performance. Hence, social network structural characteristics should be incorporated into research on new product growth as well as into managerial marketing decisions such as targeting and new product seeding.We review how social network structure influences innovations' market performance. Specifically, we discuss (1) a networks' global characteristics, namely average degree, degree distribution, clustering, and degree assortativity; (2) dyadic characteristics, or the relationships between pairs of network members, namely tie strength and embeddedness; (3) intrinsic individual characteristics, namely opinion leadership and susceptibility; and (4) location-based individual characteristics, namely the degree centrality, closeness centrality, and betweenness centrality of an individual network member.Overall, we find that growth is particularly effective in networks that demonstrate the “3 Cs”: cohesion (strong mutual influence among its members), connectedness (high number of ties), and conciseness (low redundancy). We identify gaps in current knowledge, discuss the implications on managerial decision making, and suggest topics for future research.

We analyze maximum entropy random graph ensembles with constrained degrees, drawn from arbitrary degree distributions, and a tuneable number of three-cycles (triangles). We find that such ensembles generally exhibit two transitions, a clustering and a shattering transition, separating three distinct regimes. At the clustering transition, the graphs change from typically having only isolated cycles to forming cycle clusters. At the shattering transition the graphs break up into many small cliques to achieve the desired three-cycle density. The locations of both transitions depend nontrivially on the system size. We derive a general formula for the three-cycle density in the regime of isolated cycles, for graphs with degree distributions that have finite first and second moments. For bounded degree distributions we present further analytical results on cycle densities and phase transition locations, which, while non-rigorous, are all validated via MCMC sampling simulations. We show that the shattering transition is of an entropic nature, occurring for all three-cycle density values, provided the system is large enough.

We generalize the poissonian evolving random graph model of Bauer and Bernard to deal with arbitrary degree distributions. The motivation comes from biological networks, which are well-known to exhibit non poissonian degree distribution. A node is added at each time step and is connected to the rest of the graph by oriented edges emerging from older nodes. This leads to a statistical asymmetry between incoming and outgoing edges. The law for the number of new edges at each time step is fixed but arbitrary. Thermodynamical behavior is expected when this law has a large time limit. Although (by construction) the incoming degree distributions depend on this law, this is not the case for most qualitative features concerning the size distribution of connected components, as long as the law has a finite variance. As the variance grows above 1/4, the average being <1/2, a giant component emerges, which connects a finite fraction of the vertices. Below this threshold, the distribution of component sizes decreases algebraically with a continuously varying exponent. The transition is of infinite order, in sharp contrast with the case of static graphs. The local-in-time profiles for the components of finite size allow to give a refined description of the system.

The capacity (or maximum flow) of an unicast network is known to be equal to the minimum s-t cut capacity due to the max-flow min-cut theorem. If the topology of a network (or link capacities) is dynamically changing or unknown, it is not so trivial to predict statistical properties on the maximum flow of the network. In this paper, we present a probabilistic analysis for evaluating the accumulate distribution of the minimum s-t cut capacity on random graphs. The graph ensemble treated in this paper consists of weighted graphs with arbitrary specified degree distribution. The main contribution of our work is a lower bound for the accumulate distribution of the minimum s-t cut capacity. From some computer experiments, it is observed that the lower bound derived here reflects the actual statistical behavior of the minimum s-t cut capacity of random graphs with specified degrees.

Complex networks have certain properties that characterize them. These inherent properties can be measured and applied in other fields of research. To this end, we characterize several complex networks from different domains using concepts from graph theory. In particular, the node degrees, graph spectral radius, degree assortativity, and the entire topological structure of selected complex networks are studied on the Susceptible, Infectious, Recovered (SIR) epidemic model. The results show that various complex networks properties affect the epidemic model differently. For instance nodes with high average degrees with corresponding high clustering coefficients are seen to be effective in spreading epidemics. Also the degree distribution patterns have an effect on the spreading rate of epidemic, that is dis-assortative networks are good conduits for the epidemic spreading if low degree nodes are in turn connected to high degree nodes. These results have given us a new dimension on how epidemic spreadings can be studied using complex networks as these networks possessed in them certain properties that resemble that of human society. To the best of our knowledge, this is the first work that simulated the SIR model using heterogenous complex network properties in order to study their cumulative effects on epidemic spreading.

A novel edge rewiring strategy for tuning structural properties in networks